Graph Drawing by High-Dimensional Embedding

Transcription

1 Graph Drawing by High-Dimensional Embedding David Harel and Yehuda Koren Dept. of Computer Science and Applied Mathematics The Weizmann Institute of Science, Rehovot, Israel Abstract. We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using PCA. Experiments we have carried out show the ability of the method to draw graphs of 10 5 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs. 1 Introduction A graph G(V = {1,...,n},E) is an abstract structure that is used to model a relation E over a set V of entities. Graph drawing is a standard means for the visualization of relational information, and its ultimate usefulness depends on the readability of the resulting layout; that is, the drawing algorithm s capability of conveying the meaning of the diagram quickly and clearly. Consequently, many approaches to graph drawing have been developed [2, 8]. We concentrate on the problem of drawing undirected graphs with straight-line edges, and the most popular approaches to this appear to be those that define a cost function (or a force model), whose minimization determines the optimal drawing. The resulting algorithms are known as force-directed methods. We suggest a new approach to the problem of graph drawing, relying on the observation that laying out a graph in a high dimension is significantly easier than drawing it in a low dimension. Hence, the first step of our algorithm is to quickly draw the graph in a very high dimensional space (e.g., in dimensions). Since standard visualization techniques allow using only 2 or 3 dimensions, the next step of our algorithm is to algorithmically project the high-dimensional drawing into a low dimension. For this, we have adopt a method called principal components analysis (PCA), which is well known in multivariate analysis. The resulting algorithm is extremely fast, yet very simple. Typical computation times are of less than 3 seconds for node graphs, and are thus significantly faster than force-directed approaches. As to the quality of the drawings, Section 4 shows several very encouraging results. 2 Drawing Graphs in High Dimension Frequently, drawing a graph so as to achieve a certain aesthetic criterion cannot be optimally achieved in a low dimension, due to the fact that several aesthetic goals have to compete on a shared limited space. Thus, being able to carry out the initial drawing work in many dimensions leaves more space for richer expression of the desired properties, and thus makes the entire task easier.

2 In order to draw a graph in m dimensions, we choose m pivot nodes that are almost uniformly distributed on the graph and associate each of the m axes with a unique node. Axis i, which is associated with pivot node p i, represents the graph from the viewpoint of node p i. This is done by defining the i-th coordinate of each of the other nodes as its graph-theoretic distance from p i. Hence p i is located at place 0 on axis i, its immediate neighbors are located at place 1 on this axis, and so on. More formally, denote by d uv the graph-theoretic distance between node v and node u. Let Centers be some set {p 1,p 2,...,p m } V. Each node v V is associated with m coordinates X 1 (v),x 2 (v),...,x m (v), such that X i (v) =d piv. The resulting algorithm for drawing the graph in m dimensions is given in Fig. 1. The graph theoretic distances are computed using breadth-first-search (BFS). (When edges are positively weighted, BFS should be replaced by Dijkstra s algorithm; see e.g., [1].) The set Centers is chosen as follows. The first member, p 1, is chosen at random. For j =2,...,m, node p j is a node that maximizes the shortest distance from {p 1,p 2,...,p j 1 }. This method is mentioned in [7] as a 2-approximation to the k- center problem, where we want to choose k vertices of V, such that the longest distance from V to these k centers is minimized. Function HighDimDraw (G(V = {1,...,n},E),m) % This function finds an m-dimensional layout of G: Choose node p 1 randomly from V d[1,...,n] for i =1to m do % Compute the i th coordinate using BFS: for every j V X i (j) d pij d[j] min{d[j],x i (j)} % Choose next pivot: p i+1 arg max {j V } {d[j]} return X 1,X 2,...,X m Fig. 1. Drawing a graph in m dimensions The time complexity of this algorithm is O(m E ), since we perform BFS in each of the m iterations. A typical value of m is 50. Here now are two observations regarding the properties of the resulting drawing. First, for every two nodes v and u and axis 1 i m, wehave: d uv X i (v) X i (u) This follows directly from the triangle inequality, since: d piu d piv + d uv and d piv d piu + d uv = X i (v) X i (u) = d piv d piu d uv Thus, nodes that are closely related in the graph will be drawn close together. The second observation goes in the opposite direction, and shows a kind of separation between nodes that are distant in the graph. For an axis i and nodes u and v, denote def = min{d piv,d piu}. Then, for every v, u V and axis 1 i m, wehave: δ i v,u 2

3 d uv 2δ i v,u X i (v) X i (u) For the proof, assume w.l.o.g. that δv,u i = d piv. Again, using the triangle inequality: d uv d piv + d piu = d piv + d piv +(d piu d piv) =2δv,u i + X i (v) X i (u) = d uv 2δv,u i X i (v) X i (u) Thus if we denote the minimal distance between {v, u} and Centers by: def ɛ v,u = min d p i {1,...,m},j {v,u} ij, then there exists an axis i such that X i (v) X i (u) d uv 2ɛ v,u. We note that according to the way we have chosen the pivots, we expect ɛ v,u to be fairly small. 3 Projecting Into a Low Dimension At this stage we have an m-dimensional drawing of the graph. In order to visually realize the drawing we have to project it into 2 or 3 dimensions. Picking a good projection is not straightforward, since the axes are correlated and contain redundant information. Moreover, several axes may scatter nodes better than others, thus being more informative. For example, consider a square grid (as in Fig. 3(a)). If we use two axes that correspond to two opposite corners, the resulting drawing will be essentially 1-dimensional, as the two axes convey basically the same information and are anti-correlated. (That is, being near one corner is exactly like being far the opposite corner.) Also, taking an axis associated with a boundary node is very often more informative than taking an axis associated with a central node; the first case causes the nodes to be scattered in a much better way, since the maximal distance from a boundary node is about twice as large as the maximal distance from a central node. To address these issues we use a tool that is well known and in standard use in multivariate analysis principal component analysis (PCA). PCA transforms a number of (possibly) correlated variables into a (smaller) number of uncorrelated variables called principal components (PCs). The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. By using only the first few principal components, PCA makes it possible to reduce the number of significant dimensions of the data, while maintaining the maximum possible variance thereof. See [3] for a comprehensive discussion of PCA. In our case, we have mn-dimensional variables X 1,...,X m, describing the n nodes in m dimensions. We want to represent the n nodes using only k dimensions (typically k =2), using kn-dimensional uncorrelated vectors Y 1,...,Y k, which are the principal components. Hence, the coordinates of node i are (Y 1 (i),...,y k (i)). Each of the PCs among Y 1,...,Y k is a linear combination of the original variables X 1,...,X m. def Here are the details. Denote the mean of i-th axis by m i = n X i (j) j=1 n. The first stage of the PCA is to center the data around 0 which is just a translation of the drawing. We denote the vectors of centered data by ˆX 1,..., ˆX m, defined as: ˆX i (j) =X i (j) m i, i =1,...,m, j =1,...,n 3

4 We now construct an m n matrix, X, whose rows are the (centered) coordinates: ˆX 1 (1)... ˆX1 (n) X = ˆX m (1)... ˆXm (n) The covariance matrix S, of dimension m m, is defined as S = 1 n XXT We now have to compute the first k eigenvectors of S (those that correspond to the largest eigenvalues). We denote these eigenvectors by u 1,...,u k. The vector lengths should be normalized to 1, so that these k vectors are orthonormal. A simple method for computing the eigenvectors is described below. Now to the projection itself. The first new axis, Y 1, is the projection of the data in the direction of u 1, the next axis, Y 2, is the projection in the direction of u 2, and so on. Hence the new coordinates are defined by: Y i = X T u i, i =1,...,k It can be proved that Y 1 is the 1-dimensional projection of the data that has the maximal variance (i.e., in which the data is most scattered), and also that Y 1,...,Y k constitute the k-dimensional projection of the data with the maximal variance. Moreover, these k axes are uncorrelated. In general, as we shall see in Section 4, it suffices to draw a graph on the plane using Y 1 and Y 2 only, thus scattering the nodes in a maximal fashion. 1 However, sometimes using Y 3 or Y 4 may be useful too. Regarding time complexity, the most costly step is computing the covariance matrix S = 1 n XXT. (In practice we do not divide by n, since multiplication by a constant does not change the eigenvectors.) This matrix multiplication is carried out in a straightforward way using exactly m 2 n multiplications and additions, so the time complexity is O(m 2 n), with a small constant (although matrix multiplication can be done faster in theory; see e.g., [1]). As to computing the first eigenvectors of the m m covariance matrix S (i.e., those that correspond to the largest eigenvalues), we use the simple power-iteration method; see e.g., [12]). Since m<<n, the running time is negligible (taking in practice only a few milliseconds) and there is no need for more complicated techniques. The basic idea is as follows. Say we are given an m m matrix A with eigenvectors u 1,u 2,...,u m, whose corresponding eigenvalues are λ 1 >λ 2 > >λ m. Let x R n.ifxis not orthogonal to u 1 (i.e., x u 1 0) then the series Ax, A 2 x, A 3 x,... converges in the direction of u 1. More generally, in the case where x u 1 =0,x u 2 =0,...,x u j 1 = 0,x u j 0, the series Ax, A 2 x, A 3 x,... converges in the direction of u j. The full algorithm is depicted in Fig Thus, using PCA is, in a sense, incorporating a global repulsive force, in the terms used in force-directed methods. 4

5 Function PowerIteration (S m m matrix ) % This function computes u 1,u 2,...,u k, the first k eigenvectors of S. const ɛ for i =1to k do û i random û i ûi û i do u i û i % Orthogonalize against previous eigenvectors: for j =1to i 1 do u i u i (u i u j )u j end for û i Su i û i ûi û i % normalization while û i u i < 1 ɛ u i û i end for return u 1,u 2,...,u k 4 Examples Fig. 2. The power iteration algorithm Our algorithm was implemented in C, and runs on a dual processor Intel Xeon 1.7Ghz PC. Since the implementation is non-parallel, only one of the processors is used. For all the results given here we have set m =50, meaning that the graphs are embedded in 50 dimensions. Our experience is that the results are not sensitive to the exact value of m. In fact, increasing m does not degrade the quality of the results, but doing so seems not to be needed. On the other hand, picking an overly small value for m may harm the smoothness of the drawing. Table 1 gives the actual running times of the algorithm on graphs of different sizes. In addition to the total computation time, we show the times of the two most costly parts of the algorithm computing the m-dimensional embedding (Fig. 1) and computing the covariance matrix S. We want to stress the fact that since the algorithm does not incorporate an optimization process, the running time is determined completely by the size of the graph (i.e., V and E ), and is independent of the structure of the graph. This is unlike force-directed methods. graph V E total high dim. covariance time embedding comp. matrix 516 [11] [11] elt Crack elt Sphere Sierpinski (depth 10) grid Ocean mrnga grid mrngb Taken from the University of Greenwich Graph Partitioning Archive, at: c.walshaw/partition Table 1. Running time (in seconds) of the various components of the algorithm 5

6 Graphs of around 10 5 nodes take only a few seconds to draw, and node graphs take less than a minute. Thus, the algorithm exhibits a truly significant improvement in computation time for drawing large graphs over previously known ones. Following is a collection of several drawings produced by the algorithm. The layouts shown in Fig. 3 are typical results of our algorithm, produced by taking the first two principal components as the axes. Sometimes it might be better aesthetically to take other components. For example, in Fig. 4(a) the 516 graph is depicted using the first and second principal components, while in Fig. 4(b) the first and third PCs are used, resulting in an aesthetically superior drawing. In fact, the algorithm also produces more information than others, by drawing the graph in a high dimension. Thus, we can view the graph from different viewpoints that may reveal interesting aspects of the graph. This is demonstrated in the drawing of the Sphere graph. Fig. 5(a) shows a drawing using the first and second PCs. The six smooth corners appearing in Fig. 5(a) become really salient in Fig. 5(b), where the third and forth PCs are used. These corners are still clearly visible using the forth and fifth PCs in Fig. 5(c), where a flower shape emerges. (a) (b) (c) (d) (e) (f) Fig. 3. Layouts of: (a) A grid; (b) A grid with 1 of the edges omitted at random; 3 (c) A grid with opposite corners connected; (d) A a torus; (e) Crack (f) 3elt 6

7 (a) (b) Fig. 4. Drawing the 516 graph using: (a) 1 st and 2 nd PCs; (b) 1 st and 3 rd PCs 4.1 Zooming in on regions of interest Drawings in two dimensions reveal only part of the richness of the original high dimensional drawing. Indeed, the 2D drawing must forgo showing some properties of small portions of the graph, in order to get a well balanced picture of the entire graph. This facilitates a novel kind of interactive exploration of the graph structure: The user can choose a region of interest in the drawing and ask the program to zoom in on it. We then utilize the fact that we have a high dimensional drawing of the graph, which possibly contains a better explanation for the chosen subgraph than what shows up in 2D. First we take the coordinates of the subgraph from the already computed m-dimensional drawing. We then use PCA to project these coordinates into 2D. In this way we may reveal properties appearing in the high-dimensional drawing, which are not shown in the low-dimensional drawing of the full graph. For example, we wanted to investigate the corners of the Sphere graph. We zoomed in on one of the corners, and the result is shown in Fig. 5(d). It can be seen that the corner is a meeting point of four faces. Another example is the Sierpinski fractal of depth 7, shown in Fig. 6(a). Note that the left and top parts of it are distorted. In Fig. 6(b) we depict the result of zooming-in on the left part of the graph, revealing its nice structure. The layout of the 4elt2 graph, depicted in Fig. 6(c), resembles the one obtained by [9]. For a better understanding of its structure we may zoom-in on parts of the drawing. Fig. 6(d) shows the results of zooming-in on the bottom strip. In Fig. 6(e) we provide a drawing of the Ocean graph, containing over nodes. To understand its microstructure we zoom-in on it, providing a sample result in Fig. 6(f). The last example is the 4970 graph, nicely depicted in Fig. 6(g). We zoom-in on its top-center portion, as shown in Fig. 6(h). 5 Discussion We have presented an extremely fast approach for drawing graphs. Results of the new algorithm resemble those of force-directed graph drawing algorithms. In terms of performance and simplicity the algorithm has some significant advantages when compared to force-directed methods. A simple implementation of a forcedirected method has real difficulties when dealing with graphs of more than a few hundreds of nodes. Recently, several authors, including work done in our own group, have designed multi-scale enhancements of force directed methods, which considerably improve their running time; see [5, 6, 11, 4, 10]. The fastest of them all, [11], draws a node graph in a typical time of ten minutes. Our algorithm is considerably faster, being able to draw a node graph in less than 3 seconds. Moreover, the implementation of our algorithm is much simpler and is almost parameter-free. Recently, we have designed another algorithm for drawing huge graphs, which we call Ace [9]. Ace draws a graph by quickly calculating eigenvectors of the Laplacian 7

8 (a) (b) (c) (d) Fig. 5. Multiple viewpoints of the Sphere graph: (a) first and second PCs; (b) third and forth PCs; (c) forth and fifth PCs; (d) zooming in one of the corners 8

9 (a) (b) (c) (d) (e) (f) (g) (h) Fig. 6. (a) A depth 7 Sierpinski graph; (b) zooming-in on the squeezed left side of (a); (c) the 4elt2 graph; (d) zooming-in on the bottom of (c); (e) the Ocean graph; (f) zooming-in on the micro structure of (e); (g) the 4970 graph; (h) zooming-in on the top-center portion of (g) 9

10 matrix associated with it, using a special algebraic multigrid technique. Ace can draw node graphs in about 2 seconds. However, this algorithm is quite complicated in comparison to the one described here. Moreover, the running-time of Ace (like that of force-directed methods) depends on the graph s structure, unlike our algorithm, where it depends only on the graph s size. A detailed comparison between the results of the two algorithms has yet to be done. The output of our algorithm is multi-dimensional, allowing multiple views of the graph. This also facilitates a novel technique for interactive exploration of the graph, by focusing on selected portions thereof, showing them in a way that is not possible in a 2D drawing of the entire graph. Applications to information visualization Our algorithm can deal directly with edgeweighted graphs, making it suitable for information visualization tasks. In this case it has an interesting performance-related advantage over other algorithms, including Ace: In a typical case, one is given n objects and a distance function measuring the dissimilarity between two objects. Frequently, computing the distance between two objects is a costly operation; e.g., when the objects are DNA sequences of length k, a common distance measure is the edit-distance, whose computation may take time O(k 2 ).A nice drawing puts similar objects close together, while non-similar objects are distantly located. Hence, force-directed drawing algorithms that can draw weighted graphs are suitable. However, n is typically large in such applications, so one has to consider multiscale enhancements, and these would require the computation of the close neighbors of each of the objects. This, in turn, would require the computation of the distances between all pairs, resulting in n (n 1)/2 distance computations, which is often too costly. In contrast, our method needs only m n distance computations. References 1. T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, G. Di Battista, P. Eades, R. Tamassia and I.G. Tollis, Algorithms for the Visualization of Graphs, Prentice-Hall, B. S. Everitt and G. Dunn, Applied Multivariate Data Analysis, Arnold, P. Gajer, M. T. Goodrich, and S. G. Kobourov, A Multi-dimensional Approach to Force- Directed Layouts of Large Graphs, Proceedings of Graph Drawing 2000, LNCS 1984, pp , Springer Verlag, R. Hadany and D. Harel, A Multi-Scale Method for Drawing Graphs Nicely, Discrete Applied Mathematics, 113 (2001), D. Harel and Y. Koren, A Fast Multi-Scale Method for Drawing Graphs Nicely, Proceedings of Graph Drawing 00, LNCS 1984, Springer Verlag, pp , D. S. Hochbaum (ed.), Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, M. Kaufmann and D. Wagner (Eds.), Drawing Graphs: Methods and Models, LNCS 2025, Springer Verlag, Y. Koren, L. Carmel and D. Harel ACE: A Fast Multiscale Eigenvectors Computation for Drawing Huge Graphs, Technical Report MCS01-17, The Weizmann Institute of Science, Available at: A. Quigley and P. Eades, FADE: Graph Drawing, Clustering, and Visual Abstraction, Proceedings of Graph Drawing 2000, LNCS 1984, pp , Springer Verlag, C. Walshaw, A Multilevel Algorithm for Force-Directed Graph Drawing, Proceedings of Graph Drawing 2000, LNCS 1984, pp , Springer Verlag, D. S. Watkins, Fundamentals of Matrix Computations, John Wiley,